Finite Group DatabasebetaFinite Group Databasebeta| order | 21840 |
| automorphism | Aut(G) order 25159680 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D10920 |
176 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 176 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 10920 |
A minimal faithful permutation representation — 2 generators on 36 points:
(1,2,3,4,5,6,7,8)(9,10,11)(12,13,14,15,16)(17,18,19,20,21,22,23)(24,25,26,27,28,29,30,31,32,33,34,35,36);(2,8)(3,7)(4,6)(10,11)(13,16)(14,15)(18,23)(19,22)(20,21)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)Polycyclic presentation:
2^4,3,5,7,13|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^4,g^a=g^6,h^a=h^125463 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D21840, D32760 |
| maximal normal subgroup of | D21840 |
| maximal quotient of | Dic10920, D21840, D32760 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 5, 7, 13]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^4);;
gap> SetConjugate(coll, 7, 1, F.7^6);;
gap> SetConjugate(coll, 8, 1, F.8^12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);